Problem: Solve for $x$ : $ 5|x - 7| + 4 = -2|x - 7| + 10 $
Answer: Add $ {2|x - 7|} $ to both sides: $ \begin{eqnarray} 5|x - 7| + 4 &=& -2|x - 7| + 10 \\ \\ { + 2|x - 7|} && { + 2|x - 7|} \\ \\ 7|x - 7| + 4 &=& 10 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 7|x - 7| + 4 &=& 10 \\ \\ { - 4} &=& { - 4} \\ \\ 7|x - 7| &=& 6 \end{eqnarray} $ Divide both sides by ${7}$ $ \dfrac{7|x - 7|} {{7}} = \dfrac{6} {{7}} $ Simplify: $ |x - 7| = \dfrac{6}{7}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 7 = -\dfrac{6}{7} $ or $ x - 7 = \dfrac{6}{7} $ Solve for the solution where $x - 7$ is negative: $ x - 7 = -\dfrac{6}{7} $ Add ${7}$ to both sides: $ \begin{eqnarray} x - 7 &=& -\dfrac{6}{7} \\ \\ {+ 7} && {+ 7} \\ \\ x &=& -\dfrac{6}{7} + 7 \end{eqnarray} $ Change the ${ + 7}$ to an equivalent fraction with a denominator of $7$ $ x = - \dfrac{6}{7} {+ \dfrac{49}{7}} $ $ x = \dfrac{43}{7} $ Then calculate the solution where $x - 7$ is positive: $ x - 7 = \dfrac{6}{7} $ Add ${7}$ to both sides: $ \begin{eqnarray} x - 7 &=& \dfrac{6}{7} \\ \\ {+ 7} && {+ 7} \\ \\ x &=& \dfrac{6}{7} + 7 \end{eqnarray} $ Change the ${ + 7}$ to an equivalent fraction with a denominator of $7$ $ x = \dfrac{6}{7} {+ \dfrac{49}{7}} $ $ x = \dfrac{55}{7} $ Thus, the correct answer is $x = \dfrac{43}{7} $ or $x = \dfrac{55}{7} $.